Problem: $ D = \left[\begin{array}{rr}1 & 2 \\ -2 & 3 \\ 0 & -1\end{array}\right]$ $ v = \left[\begin{array}{r}5 \\ 3\end{array}\right]$ What is $ D v$ ?
Solution: Because $ D$ has dimensions $(3\times2)$ and $ v$ has dimensions $(2\times1)$ , the answer matrix will have dimensions $(3\times1)$ $ D v = \left[\begin{array}{rr}{1} & {2} \\ {-2} & {3} \\ \color{gray}{0} & \color{gray}{-1}\end{array}\right] \left[\begin{array}{r}{5} \\ {3}\end{array}\right] = \left[\begin{array}{r}? \\ ? \\ ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ v$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ v$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ v$ , and so on. Add the products together. $ \left[\begin{array}{r}{1}\cdot{5}+{2}\cdot{3} \\ ? \\ ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ v$ and add the products together. $ \left[\begin{array}{r}{1}\cdot{5}+{2}\cdot{3} \\ {-2}\cdot{5}+{3}\cdot{3} \\ ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{r}{1}\cdot{5}+{2}\cdot{3} \\ {-2}\cdot{5}+{3}\cdot{3} \\ \color{gray}{0}\cdot{5}+\color{gray}{-1}\cdot{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{r}11 \\ -1 \\ -3\end{array}\right] $